Positive and sign-changing stationary solutions of degenerate logistic type equations

Abstract

In this work we study the existence and uniqueness of a positive, as well as a sign-changing steady-state solution of the degenerate logistic equation with a non-homogeneous superlinear term. Our outcome on a solution that changes sign, defined in higher dimensions, contribute to the existing literature of a few results for the problem, mostly developed in one dimension. We apply variational techniques, in particular the problem constrained to the Nehari manifold, and investigate how it changes as the parameter $\lambda$ in the equation or the function $b$ vary, affecting the existence and non-existence of solutions of the elliptic problem.